Something like a graph with "multi" vertices

Question

Is there a name for the following structure?

Let $U$ be a set. Let $E$ be a set of finite subsets of $U$.

I consider a labeled directed graph whose vertices are elements of $E$.

Is there a name for this?

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Note that "paths" in this structure are $p_1\dots p_n$ where $p_i$ are labels of edges and $\operatorname{Dst}p_i \subseteq \operatorname{Src}p_{i+1}$.

Answer 1

Well, $E$ is a poset ordered by $\subseteq$, because $\subseteq$ is reflexive, anti-symmetric, and transitive. Posets can be drawn as Hasse diagrams, and your $p_i$'s are the (graph) edges of a diagram.